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G = C5×C23.20D4order 320 = 26·5

Direct product of C5 and C23.20D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C5×C23.20D4, C2.D89C10, C4.Q812C10, Q8⋊C48C10, (C2×C20).462D4, C22⋊C8.5C10, C23.19(C5×D4), C22⋊Q8.7C10, (C22×C10).37D4, C10.129(C4○D8), C20.322(C4○D4), (C2×C20).941C23, (C2×C40).264C22, C42⋊C2.9C10, C22.106(D4×C10), (Q8×C10).171C22, C10.144(C8.C22), (C22×C20).433C22, C10.100(C22.D4), (C5×C4.Q8)⋊27C2, (C5×C2.D8)⋊24C2, C2.16(C5×C4○D8), C4.34(C5×C4○D4), C4⋊C4.62(C2×C10), (C2×C8).11(C2×C10), (C2×C4).108(C5×D4), (C5×Q8⋊C4)⋊36C2, (C2×C10).662(C2×D4), (C5×C22⋊C8).14C2, (C2×Q8).15(C2×C10), C2.19(C5×C8.C22), (C5×C22⋊Q8).17C2, (C5×C4⋊C4).385C22, (C22×C4).51(C2×C10), (C5×C42⋊C2).23C2, (C2×C4).116(C22×C10), C2.16(C5×C22.D4), SmallGroup(320,986)

Series: Derived Chief Lower central Upper central

C1C2×C4 — C5×C23.20D4
C1C2C4C2×C4C2×C20Q8×C10C5×C22⋊Q8 — C5×C23.20D4
C1C2C2×C4 — C5×C23.20D4
C1C2×C10C22×C20 — C5×C23.20D4

Generators and relations for C5×C23.20D4
 G = < a,b,c,d,e,f | a5=b2=c2=d2=1, e4=f2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=bc=cb, bd=db, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce3 >

Subgroups: 162 in 96 conjugacy classes, 50 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×Q8, C20, C20, C2×C10, C2×C10, C22⋊C8, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C40, C2×C20, C2×C20, C5×Q8, C22×C10, C23.20D4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C22×C20, Q8×C10, C5×C22⋊C8, C5×Q8⋊C4, C5×C4.Q8, C5×C2.D8, C5×C42⋊C2, C5×C22⋊Q8, C5×C23.20D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C2×C10, C22.D4, C4○D8, C8.C22, C5×D4, C22×C10, C23.20D4, D4×C10, C5×C4○D4, C5×C22.D4, C5×C4○D8, C5×C8.C22, C5×C23.20D4

Smallest permutation representation of C5×C23.20D4
On 160 points
Generators in S160
(1 127 159 39 151)(2 128 160 40 152)(3 121 153 33 145)(4 122 154 34 146)(5 123 155 35 147)(6 124 156 36 148)(7 125 157 37 149)(8 126 158 38 150)(9 113 137 17 129)(10 114 138 18 130)(11 115 139 19 131)(12 116 140 20 132)(13 117 141 21 133)(14 118 142 22 134)(15 119 143 23 135)(16 120 144 24 136)(25 89 53 81 41)(26 90 54 82 42)(27 91 55 83 43)(28 92 56 84 44)(29 93 49 85 45)(30 94 50 86 46)(31 95 51 87 47)(32 96 52 88 48)(57 73 109 65 101)(58 74 110 66 102)(59 75 111 67 103)(60 76 112 68 104)(61 77 105 69 97)(62 78 106 70 98)(63 79 107 71 99)(64 80 108 72 100)
(2 88)(4 82)(6 84)(8 86)(9 13)(10 101)(11 15)(12 103)(14 97)(16 99)(17 21)(18 109)(19 23)(20 111)(22 105)(24 107)(26 154)(28 156)(30 158)(32 160)(34 90)(36 92)(38 94)(40 96)(42 122)(44 124)(46 126)(48 128)(50 150)(52 152)(54 146)(56 148)(57 114)(58 62)(59 116)(60 64)(61 118)(63 120)(65 130)(66 70)(67 132)(68 72)(69 134)(71 136)(73 138)(74 78)(75 140)(76 80)(77 142)(79 144)(98 102)(100 104)(106 110)(108 112)(113 117)(115 119)(129 133)(131 135)(137 141)(139 143)
(1 87)(2 88)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 104)(10 97)(11 98)(12 99)(13 100)(14 101)(15 102)(16 103)(17 112)(18 105)(19 106)(20 107)(21 108)(22 109)(23 110)(24 111)(25 153)(26 154)(27 155)(28 156)(29 157)(30 158)(31 159)(32 160)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 149)(50 150)(51 151)(52 152)(53 145)(54 146)(55 147)(56 148)(57 118)(58 119)(59 120)(60 113)(61 114)(62 115)(63 116)(64 117)(65 134)(66 135)(67 136)(68 129)(69 130)(70 131)(71 132)(72 133)(73 142)(74 143)(75 144)(76 137)(77 138)(78 139)(79 140)(80 141)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(137 141)(138 142)(139 143)(140 144)(145 149)(146 150)(147 151)(148 152)(153 157)(154 158)(155 159)(156 160)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136)(137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152)(153 154 155 156 157 158 159 160)
(1 118 5 114)(2 60 6 64)(3 116 7 120)(4 58 8 62)(9 56 13 52)(10 151 14 147)(11 54 15 50)(12 149 16 145)(17 28 21 32)(18 159 22 155)(19 26 23 30)(20 157 24 153)(25 107 29 111)(27 105 31 109)(33 132 37 136)(34 66 38 70)(35 130 39 134)(36 72 40 68)(41 79 45 75)(42 143 46 139)(43 77 47 73)(44 141 48 137)(49 103 53 99)(51 101 55 97)(57 83 61 87)(59 81 63 85)(65 91 69 95)(67 89 71 93)(74 126 78 122)(76 124 80 128)(82 119 86 115)(84 117 88 113)(90 135 94 131)(92 133 96 129)(98 146 102 150)(100 152 104 148)(106 154 110 158)(108 160 112 156)(121 140 125 144)(123 138 127 142)

G:=sub<Sym(160)| (1,127,159,39,151)(2,128,160,40,152)(3,121,153,33,145)(4,122,154,34,146)(5,123,155,35,147)(6,124,156,36,148)(7,125,157,37,149)(8,126,158,38,150)(9,113,137,17,129)(10,114,138,18,130)(11,115,139,19,131)(12,116,140,20,132)(13,117,141,21,133)(14,118,142,22,134)(15,119,143,23,135)(16,120,144,24,136)(25,89,53,81,41)(26,90,54,82,42)(27,91,55,83,43)(28,92,56,84,44)(29,93,49,85,45)(30,94,50,86,46)(31,95,51,87,47)(32,96,52,88,48)(57,73,109,65,101)(58,74,110,66,102)(59,75,111,67,103)(60,76,112,68,104)(61,77,105,69,97)(62,78,106,70,98)(63,79,107,71,99)(64,80,108,72,100), (2,88)(4,82)(6,84)(8,86)(9,13)(10,101)(11,15)(12,103)(14,97)(16,99)(17,21)(18,109)(19,23)(20,111)(22,105)(24,107)(26,154)(28,156)(30,158)(32,160)(34,90)(36,92)(38,94)(40,96)(42,122)(44,124)(46,126)(48,128)(50,150)(52,152)(54,146)(56,148)(57,114)(58,62)(59,116)(60,64)(61,118)(63,120)(65,130)(66,70)(67,132)(68,72)(69,134)(71,136)(73,138)(74,78)(75,140)(76,80)(77,142)(79,144)(98,102)(100,104)(106,110)(108,112)(113,117)(115,119)(129,133)(131,135)(137,141)(139,143), (1,87)(2,88)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,104)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,112)(18,105)(19,106)(20,107)(21,108)(22,109)(23,110)(24,111)(25,153)(26,154)(27,155)(28,156)(29,157)(30,158)(31,159)(32,160)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,149)(50,150)(51,151)(52,152)(53,145)(54,146)(55,147)(56,148)(57,118)(58,119)(59,120)(60,113)(61,114)(62,115)(63,116)(64,117)(65,134)(66,135)(67,136)(68,129)(69,130)(70,131)(71,132)(72,133)(73,142)(74,143)(75,144)(76,137)(77,138)(78,139)(79,140)(80,141), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160), (1,118,5,114)(2,60,6,64)(3,116,7,120)(4,58,8,62)(9,56,13,52)(10,151,14,147)(11,54,15,50)(12,149,16,145)(17,28,21,32)(18,159,22,155)(19,26,23,30)(20,157,24,153)(25,107,29,111)(27,105,31,109)(33,132,37,136)(34,66,38,70)(35,130,39,134)(36,72,40,68)(41,79,45,75)(42,143,46,139)(43,77,47,73)(44,141,48,137)(49,103,53,99)(51,101,55,97)(57,83,61,87)(59,81,63,85)(65,91,69,95)(67,89,71,93)(74,126,78,122)(76,124,80,128)(82,119,86,115)(84,117,88,113)(90,135,94,131)(92,133,96,129)(98,146,102,150)(100,152,104,148)(106,154,110,158)(108,160,112,156)(121,140,125,144)(123,138,127,142)>;

G:=Group( (1,127,159,39,151)(2,128,160,40,152)(3,121,153,33,145)(4,122,154,34,146)(5,123,155,35,147)(6,124,156,36,148)(7,125,157,37,149)(8,126,158,38,150)(9,113,137,17,129)(10,114,138,18,130)(11,115,139,19,131)(12,116,140,20,132)(13,117,141,21,133)(14,118,142,22,134)(15,119,143,23,135)(16,120,144,24,136)(25,89,53,81,41)(26,90,54,82,42)(27,91,55,83,43)(28,92,56,84,44)(29,93,49,85,45)(30,94,50,86,46)(31,95,51,87,47)(32,96,52,88,48)(57,73,109,65,101)(58,74,110,66,102)(59,75,111,67,103)(60,76,112,68,104)(61,77,105,69,97)(62,78,106,70,98)(63,79,107,71,99)(64,80,108,72,100), (2,88)(4,82)(6,84)(8,86)(9,13)(10,101)(11,15)(12,103)(14,97)(16,99)(17,21)(18,109)(19,23)(20,111)(22,105)(24,107)(26,154)(28,156)(30,158)(32,160)(34,90)(36,92)(38,94)(40,96)(42,122)(44,124)(46,126)(48,128)(50,150)(52,152)(54,146)(56,148)(57,114)(58,62)(59,116)(60,64)(61,118)(63,120)(65,130)(66,70)(67,132)(68,72)(69,134)(71,136)(73,138)(74,78)(75,140)(76,80)(77,142)(79,144)(98,102)(100,104)(106,110)(108,112)(113,117)(115,119)(129,133)(131,135)(137,141)(139,143), (1,87)(2,88)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,104)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,112)(18,105)(19,106)(20,107)(21,108)(22,109)(23,110)(24,111)(25,153)(26,154)(27,155)(28,156)(29,157)(30,158)(31,159)(32,160)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,149)(50,150)(51,151)(52,152)(53,145)(54,146)(55,147)(56,148)(57,118)(58,119)(59,120)(60,113)(61,114)(62,115)(63,116)(64,117)(65,134)(66,135)(67,136)(68,129)(69,130)(70,131)(71,132)(72,133)(73,142)(74,143)(75,144)(76,137)(77,138)(78,139)(79,140)(80,141), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160), (1,118,5,114)(2,60,6,64)(3,116,7,120)(4,58,8,62)(9,56,13,52)(10,151,14,147)(11,54,15,50)(12,149,16,145)(17,28,21,32)(18,159,22,155)(19,26,23,30)(20,157,24,153)(25,107,29,111)(27,105,31,109)(33,132,37,136)(34,66,38,70)(35,130,39,134)(36,72,40,68)(41,79,45,75)(42,143,46,139)(43,77,47,73)(44,141,48,137)(49,103,53,99)(51,101,55,97)(57,83,61,87)(59,81,63,85)(65,91,69,95)(67,89,71,93)(74,126,78,122)(76,124,80,128)(82,119,86,115)(84,117,88,113)(90,135,94,131)(92,133,96,129)(98,146,102,150)(100,152,104,148)(106,154,110,158)(108,160,112,156)(121,140,125,144)(123,138,127,142) );

G=PermutationGroup([[(1,127,159,39,151),(2,128,160,40,152),(3,121,153,33,145),(4,122,154,34,146),(5,123,155,35,147),(6,124,156,36,148),(7,125,157,37,149),(8,126,158,38,150),(9,113,137,17,129),(10,114,138,18,130),(11,115,139,19,131),(12,116,140,20,132),(13,117,141,21,133),(14,118,142,22,134),(15,119,143,23,135),(16,120,144,24,136),(25,89,53,81,41),(26,90,54,82,42),(27,91,55,83,43),(28,92,56,84,44),(29,93,49,85,45),(30,94,50,86,46),(31,95,51,87,47),(32,96,52,88,48),(57,73,109,65,101),(58,74,110,66,102),(59,75,111,67,103),(60,76,112,68,104),(61,77,105,69,97),(62,78,106,70,98),(63,79,107,71,99),(64,80,108,72,100)], [(2,88),(4,82),(6,84),(8,86),(9,13),(10,101),(11,15),(12,103),(14,97),(16,99),(17,21),(18,109),(19,23),(20,111),(22,105),(24,107),(26,154),(28,156),(30,158),(32,160),(34,90),(36,92),(38,94),(40,96),(42,122),(44,124),(46,126),(48,128),(50,150),(52,152),(54,146),(56,148),(57,114),(58,62),(59,116),(60,64),(61,118),(63,120),(65,130),(66,70),(67,132),(68,72),(69,134),(71,136),(73,138),(74,78),(75,140),(76,80),(77,142),(79,144),(98,102),(100,104),(106,110),(108,112),(113,117),(115,119),(129,133),(131,135),(137,141),(139,143)], [(1,87),(2,88),(3,81),(4,82),(5,83),(6,84),(7,85),(8,86),(9,104),(10,97),(11,98),(12,99),(13,100),(14,101),(15,102),(16,103),(17,112),(18,105),(19,106),(20,107),(21,108),(22,109),(23,110),(24,111),(25,153),(26,154),(27,155),(28,156),(29,157),(30,158),(31,159),(32,160),(33,89),(34,90),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,121),(42,122),(43,123),(44,124),(45,125),(46,126),(47,127),(48,128),(49,149),(50,150),(51,151),(52,152),(53,145),(54,146),(55,147),(56,148),(57,118),(58,119),(59,120),(60,113),(61,114),(62,115),(63,116),(64,117),(65,134),(66,135),(67,136),(68,129),(69,130),(70,131),(71,132),(72,133),(73,142),(74,143),(75,144),(76,137),(77,138),(78,139),(79,140),(80,141)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(105,109),(106,110),(107,111),(108,112),(113,117),(114,118),(115,119),(116,120),(121,125),(122,126),(123,127),(124,128),(129,133),(130,134),(131,135),(132,136),(137,141),(138,142),(139,143),(140,144),(145,149),(146,150),(147,151),(148,152),(153,157),(154,158),(155,159),(156,160)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136),(137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152),(153,154,155,156,157,158,159,160)], [(1,118,5,114),(2,60,6,64),(3,116,7,120),(4,58,8,62),(9,56,13,52),(10,151,14,147),(11,54,15,50),(12,149,16,145),(17,28,21,32),(18,159,22,155),(19,26,23,30),(20,157,24,153),(25,107,29,111),(27,105,31,109),(33,132,37,136),(34,66,38,70),(35,130,39,134),(36,72,40,68),(41,79,45,75),(42,143,46,139),(43,77,47,73),(44,141,48,137),(49,103,53,99),(51,101,55,97),(57,83,61,87),(59,81,63,85),(65,91,69,95),(67,89,71,93),(74,126,78,122),(76,124,80,128),(82,119,86,115),(84,117,88,113),(90,135,94,131),(92,133,96,129),(98,146,102,150),(100,152,104,148),(106,154,110,158),(108,160,112,156),(121,140,125,144),(123,138,127,142)]])

95 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I4J5A5B5C5D8A8B8C8D10A···10L10M10N10O10P20A···20P20Q···20AF20AG···20AN40A···40P
order1222244444444445555888810···101010101020···2020···2020···2040···40
size111142222444488111144441···144442···24···48···84···4

95 irreducible representations

dim111111111111112222222244
type+++++++++-
imageC1C2C2C2C2C2C2C5C10C10C10C10C10C10D4D4C4○D4C4○D8C5×D4C5×D4C5×C4○D4C5×C4○D8C8.C22C5×C8.C22
kernelC5×C23.20D4C5×C22⋊C8C5×Q8⋊C4C5×C4.Q8C5×C2.D8C5×C42⋊C2C5×C22⋊Q8C23.20D4C22⋊C8Q8⋊C4C4.Q8C2.D8C42⋊C2C22⋊Q8C2×C20C22×C10C20C10C2×C4C23C4C2C10C2
# reps11211114484444114444161614

Matrix representation of C5×C23.20D4 in GL4(𝔽41) generated by

37000
03700
00370
00037
,
1100
04000
0010
00440
,
1000
0100
00400
00040
,
40000
04000
0010
0001
,
381200
01400
00518
003536
,
381200
6300
00439
002837
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,37,0,0,0,0,37],[1,0,0,0,1,40,0,0,0,0,1,4,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[38,0,0,0,12,14,0,0,0,0,5,35,0,0,18,36],[38,6,0,0,12,3,0,0,0,0,4,28,0,0,39,37] >;

C5×C23.20D4 in GAP, Magma, Sage, TeX

C_5\times C_2^3._{20}D_4
% in TeX

G:=Group("C5xC2^3.20D4");
// GroupNames label

G:=SmallGroup(320,986);
// by ID

G=gap.SmallGroup(320,986);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1128,1766,226,10085,2539,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=1,e^4=f^2=d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,e*b*e^-1=b*c=c*b,b*d=d*b,f*b*f^-1=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^3>;
// generators/relations

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